Question

Student Debt – Vermont: You take a random sample of 31 college students in the state of Vermont and find the mean debt is $25,000 with a standard deviation of $2,700. We want to construct a 90% confidence interval for the mean debt for all Vermont college students.

(a) What is the point estimate for the mean debt of all Vermont college students?
$
(b) What is the critical value of t for a 90% confidence interval? Use the value from the t-table.

(c) What is the margin of error for a 90% confidence interval? Round your answer to the nearest whole dollar.
$
(d) Construct the 90% confidence interval for the mean debt of all Vermont college students. Round your answers to the nearest whole dollar.
(  ,  )

(e) Interpret the confidence interval.

A We expect that 90% of all Vermont college students have a debt that's in the interval.

B We are 90% confident that the mean student debt of all Vermont college students is in the interval.

C We are confident that 90% of all Vermont college students have a debt that's in the interval.

D We are 10% confident that the mean student debt of Vermont college students is in the interval.

(f) We are never told whether or not the parent population is normally distributed. Why could we use the above method to find the confidence interval?

A Because the sample size is greater than or equal to 30.Because the sample size is greater than or equal to 15.   

B Because the margin of error is positive.Because the margin of error is less than or equal 30.

C Because the margin of error is positive.

D Because the margin of error is less than or equal 30.

Answer 1

Solution :

Given that,

a) Point estimate = sample mean = = $25000

sample standard deviation = s = $2700

sample size = n = 31

Degrees of freedom = df = n - 1 = 31 - 1 = 30

b) At 90% confidence level

= 1 - 90%

=1 - 0.90 =0.10

/2 = 0.05

t/2,df = t_0.05,30 = 1.697

c) Margin of error = E = t_/2,df * (s /n)

=1.697 * (2700 / 31)

Margin of error = E = $823

d) The 90% confidence interval estimate of the population mean is,

  ± E  

= $ 25000  ± $ 823

= ($24177, $25823)

f) A Because the sample size is greater than or equal to 30.Because the sample size is greater than or equal to 15.